Many-Valued Models

Many-valued models, besides providing a natural semantical interpretation for several non-classical logics, constitute a very sharp tool for investigating and understanding meta-logical properties in general. Although open to debates from the philosophical perspective, seen from the mathematical viewpoint many-valued matrices and algebras are perfectly well-defined mathematical objects with several attractive properties. This tutorial intends to review the main results, techniques and methods concerning the application of the many-valued approach to logic as a whole. 1 On many-valued thinking The technique of using finite models defined by means of tables (which turns out to be finite algebras) is arguably older than many-valued logics themselves, and has provided much information not only about non-classical systems as relevant logics, linear logic, intuitionistic logics and paraconsistent logics, but also about fragments of classical logic. The problem of enumerating such algebras satisfying given constraints is an interesting general problem and has received attention from several different areas. In this tutorial we present an elementary but general approach on small finite models, showing their relevance and reviewing some elementary methods and techniques on their uses. There are many significant names in the history of logic that are connected with the idea of many-valuedness, for different reasons. The Polish logician and philosopher Jan Łukasiewicz was born in Łvov. His philosophical work developed around on mathematical logic; Łukasiewicz dedicated much attention to many-valued logics, including his own hierarchy of many-valued propositional calculus, considered to be the first non-classical logical calculus. He is also responsible for an elegant axiomatizations of classical propositional logic; it has just three axioms and is one of the most used axiomatizations today: